A linear function can refer to two slightly different concepts. In geometery and elementary algebra a linear function is a first degree polynomial mathematical function of the form:

f(x) = m x + c

where m and c are constants.

The problem with this geometric definition is that functions of the above form - despite their names - do not necessarily satisfy the conditions of a linear map. Therefore, some people refer to functions of the above form as affine functions. If and only if a function is of the above form with c equal to zero, the function satisfies the properties of a linear map, preserving scalar multiplication and vector addition for all points in its domain.

Linear functions always have as domain the set of all real numbers and a range of all real numbers. By the geometric definition, the derivative of linear functions in terms of their independent variable, x, is always the constant m.

Linear functions (according to the geometric definition) can also be written in the form:

y = m x + c

and plotted on an X-Y-graph. It forms a straight line, as the name implies.

The constant m is often called the slope or gradient while c is the y-intercept, which gives the point of intersection between the graph of the function and the y-axis.

Examples:

• f(x)= 2x + 1

• f(x) = x

• f(x)= 9 x - 2
• f(x)= -3 x + 4

On a line graph, changing m makes the line steeper or shallower, and changing c moves the line up or down.

As mentioned, the line crosses the y-axis at the co-ordinate (0,c). It crosses the x-axis at (-c / m) (solving for 0 = m x + c we get x = -c / m).

Polynomials

Lineare Funktion | Линейна функция | Lineare Funktion | Función lineal | Lineaarinen funktio | פונקציה לינארית | funzione lineare | 一次関数 | Lineaire functie | Funkcja liniowa | Função linear | Линейная функция | Linearna funkcija | Линеарна функција